VI ma Han A=(a n) khong suy men, nen co ma tran nghich dge 11 = A ' = (bd.
Taco non vi r= v*.
ieI
Trong khi do Vi° F {f E I f(e) = 0 V, g(x).
Do do V ic' la khong gian con sinh bai f, (NhAc lai f, (e) = 6,j — 'hi 2.26) ma EV? x V* (bai 2.26).
idl
2.28. c) Gia sit {6} i E I la ed so vo han dm V; f, e V* sao cho , (e) = §i
T, la khong gian con sinh btli {f1}, T, V* (Kern bai 2.26). f e V* sao cho Re) = 1 vOi moi i E I, xet T2 la khong gian inh bai f T2 (1 T1=101 (Ty n T1P = V; nhung T 10 =401 cV.
T2° t V vi T 2° chi chita cac MAI) ti) El:t i e, ma Ea ; =0.
lei lei
Slut very (T2 (1 T1) ° a V+ T20 .
2.30. Ta c6 p la phep chigu .(r). p 2 = p <=> Id — p — p + p' = Id — 2P + p 2 = Id — p
(Id — p) 2 = (Id — p).;=, Id — p la phep chigu. Bay gib n6u y e Imp thi y= p(x) (Id — p) (y) = 0
y e Ker (Id — p).
Node lai, n6u y e Ker (Id — p) (Id — p) (y) = 0 p(y) = y= y e Imp.
Nhti very, neM p la phep chi6u thi Imp = Ker (Id — p). D6 y rang p = Id - (Id — p), to co Kerp = Im (Id — p). 2.32. a) trang hip u = 0 la tam thtlang.
GiO sit {e l , ..., es} la cd sd cim Imu. Khi do en e, (i = 1, s) sao cho u (e,)= c i . Ta có dim (Keru) = n — s. LO {e sti , e„} la co sa dm V (xem bai tap 2.3). Chi can LIS sung de" co co sä ..., E s ,
£ 0 } Oil to dude hai cd sa can tim.
b) Xet v e End (V) ma v(c) = e„ i = n thi you la phop chi6u ciia V len khong gian sinh cj song song voi khong gian con sinh btli {e„„ e„}.
§ 2 HE PHUT$NG TRiNH TUYEN TiNH 1 -7 4 2 0' 2.34. A = 2 -5 3 2 1 rang A = 2. , 5 -8 5 4 3 4 -1 1 2 3,
Nhu vdy khong gian nghiem co so" chiOu 3. FLO da cho tUdng dudng vOi
7x 2 + 4x 3 + 2x4 = 2x 1 5x 2 + 3x 3 + 2x 4 + x 5 = 0 1 4 7 xi 9 — X 4 9 9x5. 5 2 1 X 2 = 9 — 9 X5 + — 9 X4 — — ' Xr . '
Ta có mot he nghiem cd bin, chAng han la: Et i =(4, 2, 3, 3, 3)
a2 = (-7 , -1, 0, 0, 9)
2.35. HD. He c6 nghiem khong tam thang khi va chi khi ih thiic cua he Wang khong a X = 35. Khi d6 hang Mitt ma .n cite he s6 Wang 3 a khong gian nghiem 1 thigh.
2.36. HD. Coi he da cho la he phudng trinh thudn ntat viii biers (It - c)x, (c - a)y, (a - b)z. Ta dan Mit he thuc:
(I) (c-a)y -b)z 1 bz -cy cz - ay ay -bx k
De he c6 nghiem khong tam thitang, to phai c6 k = ±1. 2.37. = 1 , 4 a + b + c + 1 x2 =- 4(a-b+c+d) x4 = 4 —1 + b + c-d) 2.38. A= 2 1 3 1 m I -1 1 -m , det A = 2m (2-m). a) v6i m* 0 va m # 2 a He la he Cramer. b) m = 0 rang A = 2 vi 2 1 # 0 . 1 0
=0. 1 1 1 1 Yt y2 32 3 Y4
Digu kien có nghi5m la
2 1 a 1 0 b 1 0 b 3 1 c =a+ b - c = 0 . x=b-z Khi do {,y ,=a-2b+3z, z my c) m = 2 =4 rang A = 0.
Digu kien có nghiem: -5a + b + 3c = 0
Khi do 2a -b x -z + 3 y- z+ 2b 3 22 x 2 +y 2 x 2 2 2 X3 + Y3 2 2 x4 +Y4 2.39. PS: x3 x4 2.40, f(x) = ax 3 + bx2 + cx + d 0-1) = -a+b-c+d = 0 1(1) = a+b+c+d = 4 f(2) = 8a+4b+2c+d = 3 03) = 27a+ 914+3c+d = 16 Tito: a = 2 b = -5, c = 0 , d = 7 f(x) 2x3 - 5x2 + 7 2.41. Ta có A = (a 4), nen Za dA ii = 0 (1) (i = 1, 2, ..., n) 1=1 (adsbygoogle = window.adsbygoogle || []).push({});
Do A„ # 0, nen rang A = n-1 va khong gian nghiOni cna (1) co so. chigu 1. Hay rang (A11) = 1.
TV do hO EA ii x i =0 (i = 1, 2, n) Wong doing voi j=1
X2 X2 ± Al2X, = O.
Ta có mot he nghiOm co ban, riling ban = (-Al2 , A 11 , 0, ... n2= 0. An,
0) 0)
Et„_i =(-Alloo,o, A„)
§3. CA.0 TRUC CUA MOT TV DoNG CAU
2.42. Gia s& u la Mt chigu u2 = u. Ngu x la vec to rioting ung voi gia tri rieng ?. u(x) = Ax.
u2(x) = u(x) = Xu(x) = X2x. Nhung u2 = u nen: A 2x = Xx (1.2 - ),.)x = 0. Do x s 0 nen - X = 0 X= 0 hoar = 1.
Tudng to del vdi phep doi hop v2 = Id. 2.43. eau a), b) dO dang.
c) Dao ciia a) khong dung. Vi du:
Cho A = In (ma tran don vi), B = ma (On tam gihc (Judi ma the phan t& dgu bang 1 Khi do det (A-AIn) = det (B-AIn) = (1 - Xr. Nhung A AM B khong &Ong dang, vi A chi ding dung vii chinh no (xem bai 2.18).
Dao cua b) cling khong dung. Vi du: Chan Ala ma trar (cap n > 1) va B e Mat (n, It); B= (b id ma b 11 = 1, cent; = 0 mm (I, j) # (1, 1).
Khi do det A = det B = 0, nhung det (A - XIn) = (-k)° con I (B - Mn) = (1 - A) (-X)11-1 .
2.44.Vi N e End V, trong cd sa nao do co ma tram A nen thtic dac trung cUa N co
IA - XIn I = (-X)" + C1(-X) 111 + + C.
De thgy C„ = detA (cho X = 0). Ck 11 he so cim cac se; hang
s6 mu (-X)^4, (Woe tao thAnh bai tich cua n-k phan to tr
&fang cheo chinh dang: (a yi - X)... (a ind,,,, -X) vdi cac phgn trong khai trie'n dinh thtic IA - I, khOng chVa T. va ni trong cac dang, cot ba vdi cac dong, cOt c6 chi se; {i 1 Do dinh thtic con chinh cap k ciia A co cac phgn tv tren &tang ch chinh veil chi so" ba .•• VI 60 {II ..• Lk} la thy $, nen
s6 cua (4)"4 chinh bang te'ng tat ca cac dinh thuic con chi dang tren.
2.45. Ccich 1. Gia su A la ma tran cila cp trong cd sa nao dm V, thi = rang (9 - XDId) = rang (A- koI). CIA sii X la vac cOt, X thuoc khOng gian rieng P, ling vdi gia trt rieng X 0 khi chi khi ( A - A01)X = 0 > dim P kn = n-r. Goi R 1 la kitting git 0 rieng suy Ong ling vdi gia tri rieng Xo, thi dim E xci = p (130i cl ko), ma Pxo c R ko nen n - r < p. Mat khac do det I A-X.011=0, ni ran= 1 S. n-r. Vay 1 S n-r p.
Cdch 2. Dat B = A - kJ. Khi d6 = I A - (p+X, D)I I = A - XI I , voi A = µ + X. Nhd vay A = Xo la nghiem bet p cua IA - XI I = 0 thi µ= 0 la nghiem bei p cna. IB-µII = 0. Theo bid 2.44, I B -µII = (-0" + C1( -0"1 + + Ck (- 11) "a C, a
do Ck 11 tong cga cac Binh tilde con chinh cap k dia. B. Do rang B = r nen cac dinh there con chinh cap k > r dgu triet lieu Ck = 0 voi moi k> r. Vey I B - I = (-P)"+ + Doclop=n-s?.n-r. 2i 2.46. DS. Ak = 21 cos k n +1 C. (-1,)" 5, s < r.
HD. Xem bai 1.19, cam a) dat a + p = -A, a.p = -1.
2.47. Goi p(x) le da tilde dac tning cga ma trap A.
-A 1 G '
-A 1
an-1 an-2
thing coot to cg:
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P(A)=(-1)""a2,2det :v 1 +(-1)"*2 a,, 2 det
P(x) = det Khai trim theo
o' 1, f- k 0 0' 1) (-1)2" (a. - A) det
Nhu v0y:
P(X) = (-1)' (a._, + a„. 2 X + + asXn- ') + (- 1)11 . l. n . Ke't qua tiep theo suy ra tit ceng dr& nay.
2.49. HD: Phan Hell a thanh tich cac yang xfch doe 10 a = ... T2 . T I ; gia {e1 , ..., en} la cd s6; tn nhien cua Ca, tt u(e) =e a . Sap x6p lai co sa fe;} theo thft ty thich hdp ta dude c sa mdi cua C". sap thanh P nhom, m6i nh6m g6m m a vet t (mk bang de dai Tk), k = 1, p. Chang han nhom thit nhat gen fe' l , e',„ 1} thl u (e';) = e't+1, u(e'nil) = e' I . Tuong t>i cho ca
nho khac. v0y. Da flak cl0c trung cua u co clang I A - kin I= D I ... D„ a do Dk la da thile b•c m k dang:
- x 0 0 1
1 -X 0 0
Dk = 0 1 - X =(—OnIk+I-P(
- Ark
0 0 0 1-1.
= E lrk ()Link Nhu vay